Slides KTH stability and transition
Transcript of Slides KTH stability and transition
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Non modal stability
!"#$ &'$()* + ,-*-' .#/01)
!1((2 3!45 6-(*'-7 89: ;-#/$(1#-'1$? 6@??-A- !@()@(7 B8
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4"*?1(-
.*$C1?1*D @E F"1)
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$( -H$0>?- @E 1(
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P"0-'1#$?
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61''"< #?@")< )-J-?@>1(A 1( $ L-* $#- ./"V?- RPK.KT
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K( -H$0>?- @E 8-?J1(M:-?0@?*W
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9'$(1>- F@N -H>-'10-(* RXYYZT
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9'$(1>- F@N -H>-'10-(* RXYYZT
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9'$(
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6?$
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&D>$
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4-( F@N
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9N@ #@(#->*< @E
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.*$C1?1*D $($?D
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:D)'@)D($01#
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.*$C1?1*D
.*$C?- P-"*'$?
B(
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4J-'J1-N
.*$C1?1*D $($?D
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.*$C1?1*D $($?D'@C?-0
U = (U , V , W ) , T =T(xi)
u
t= F(u;U)
u
t = A(U)u
u(xi, t) = u eit
iu= A(U)u
u(xi, t), T(xi, t)
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!1(-$' 0 i = 0 i < 0
B(
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9/-'0$? 1(
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9-'01
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9-'01$(
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x or y
z
T0
T0 T
d
2(0&"%3456"-(.7 %-/'(*%&%'0
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2(0&"%3456"-(.7 %-/'(*%&%'0
Linear stability theory: above a critical Rayleigh number of
1708 the conductive state becomes unstable to infinitesimal
perturbations
Energy stability theory: below a critical Rayleigh number of
1708 finite-amplitude perturbations superimposed on the
conductive state decay monotonically in energy
Rayleighnumber: ratio between buoyancy forces (temperaturegradient) and viscous forces the governing parameter
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2(0&"%3456"-(.7 %-/'(*%&%'0
Linear stability theory: above a critical Rayleigh number of
1708 the conductive state becomes unstable to infinitesimal
perturbations
Energy stability theory: below a critical Rayleigh number of
1708 finite-amplitude perturbations superimposed on the
conductive state decay monotonically in energy
Rayleighnumber: ratio between buoyancy forces (temperaturegradient) and viscous forces the governing parameter
89:".%1"-'/ /4+$ '4" +-/"' +;
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C&(-" C+%/",%&&" D+$
Reynoldsnumber: ratio between inertial forces and viscous forces thegoverning parameter
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C&(-" C+%/",%&&" D+$
Linear stability theory: above a critical Reynolds number of
5772 the parabolic profile becomes unstable to infinitesimal
perturbations
Energy stability theory: below a critical Reynolds number of
49.6 finite-amplitude perturbations superimposed on the
parabolic profile decay monotonically in energy
Reynoldsnumber: ratio between inertial forces and viscous forces thegoverning parameter
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C&(-" C+%/",%&&" D+$
Linear stability theory: above a critical Reynolds number of
5772 the parabolic profile becomes unstable to infinitesimal
perturbations
Energy stability theory: below a critical Reynolds number of
49.6 finite-amplitude perturbations superimposed on the
parabolic profile decay monotonically in energy
Reynoldsnumber: ratio between inertial forces and viscous forces thegoverning parameter
89:".%1"-'/ /4+$ ',.*,&"-' :('
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9N@ @>>@
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Energy equation
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Linear growth mechanisms
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."C#'1G#$? *'$(
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."C#'1G#$? *'$(
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."C#'1G#$? *'$(
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P@(M0@)$? $>>'@$#/
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!1(-$' '@C?-0
.*$'* E'@0 ?1(-$'1@(-(G$?
U
d
dtq = Lq
q= exp(tL)q0; q(t= 0) = q0
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P@'0 @E 0$*'1H -H>@(-(G$?
=(>"* @"*>"* $($?D
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P@'0 @E 0$*'1H -H>@(-(G$?
=(>"* @"*>"* $($?D
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;$*'1H (@'0
m"#?1)-$(
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m1A-(J$?"-< J< ,'@>$A$*@'P@'0
;$*'1H -H>@(-(G$? )1l#"?* *@ #@0>"*-
.D
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m1A-(J$?"-< J< ,'@>$A$*@'P@'0
B>>-' $() ?@N-' C@"()< @E
Lower bound
G(t)
e2tmax || exp(tL)||2
|| exp(tL)||2 =|| exp(tSS1)||2
||S||2||S1||2e2tmaxUpper bound
9/- -(-'AD #$((@* )-#$D $* $ E$
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&@"()< @E */- 0$*'1H -H>@(-(G$?
e2tmax || exp(tL)||2 ||S||2||S1||2e2tmax
Two distinct cases:
(S) =||S||2||S1||2Condition number:
(S
) = 1
,::". (-7 &+$". *+,-7
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&@"()< @E */- 0$*'1H -H>@(-(G$?
e2tmax || exp(tL)||2 ||S||2||S1||2e2tmax
Two distinct cases:
(S) =||S||2||S1||2Condition number:
,::". (-7 &+$". *+,-7
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P@(M(@'0$?1*D
(S) =||S||2||S1||2 = 1
(S) =||S||2
||S1
||2
1
H+.1(& /'(*%&%'0 :.+*&"1E
@'*/@A@($? -1A-(J-#*@'$
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6?$
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&D>$
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4>G0$? 1(1G$? #@()1G@(
Initial condition that results in the maximumenergyamplification at a given time
||q(t)||2 =||q0||2 = 1
q(t) = exp(tL)q0
exp(tL) q0 = || exp(tL)|| q(t)
>'@>$A$*@' 1(>"* $0>?1]#$G@( @"*>"*
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4>G0$? 1(1G$? #@()1G@(
exp(t
L) q0 = || exp(t
L)|| q(t
)
>'@>$A$*@' 1(>"* $0>?1]#$G@( @"*>"*
Singular value decomposition of a matrix
V = UA = UVH
Au1v1
1 = ||A||2
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4>G0$? 1(1G$? #@()1G@(
exp(t
L) q0 = || exp(t
L)|| q(t
)
>'@>$A$*@' 1(>"* $0>?1]#$G@( @"*>"*
Singular value decomposition of a matrix
u1v1
svd(exp(tL)) = UVH
exp(tL)
G(t) =|| exp(tL)||
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4>G0$? 1(1G$? #@()1G@(
exp(t
L) q0 = || exp(t
L)|| q(t
)
>'@>$A$*@' 1(>"* $0>?1]#$G@( @"*>"*
Singular value decomposition of a matrix
u1v1
svd(exp(tL)) = UVH
exp(tL)
G(t) =|| exp(tL)||
Optimal initial conditionleft principal
singular vector
Optimal final conditionright principalsingular vector
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4>G0$? )1
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Lift-upmechanism in shear layers(Ellingsen & Palm 1975, Landahl 1980)
Particle displaced in thewall-normal directionretain their horizontal
momentum
Streamwise vorticesinduce streamwise
streaks
In boundary layers: wall-normal shear is O(Re)
streak growth O(Re)
Streamwisevortices
Streamwisestreaks
P@(M;@)$? ^'@N*/
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4"*?1(-
.*$C1?1*D @E F"1)
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I-#->GJ1*D
=(*-'-
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4>G0$? '-@(
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&@"()< @E '-
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&@"()< @E '-
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&@"()< @E '-
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4>G0$? E@'#1(A
.1(A"?$' J$?"- )-#@0>@"*=(>"*
||qp|| =|| f|| = 1
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I-
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I-
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6@0>@(-(*MN1
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6@0>@(-(*MN1"*M@"*>"* $($?D
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K)L@1(*
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K)L@1(*
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mH>$(
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BGJ1*D
ut
+L(U,Re)u+p= 0
u = 0
Linearized Navier-Stokes(u, p)
(f+
,m
+
)Differentiable fields
0 = u
t
+L(U,Re)u+pf+ + ( u)m+Sum and multiply
Integrate by parts overtime and space
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BGJ1*D
Integrate by parts over time and space
L(U,Re)u= U u+ u U 1
Re
2u
L+
(U,Re)f+
=U
f+
U
f+
+
1
Re2
f+
J=U(uf+) + 1
Re(f+ u uf+) +m+u+pf+
where
Z0
ZD
u
t +L(U,Re)u+p
f+ + ( u)m+
=
Z t
0
ZD
u
f+
t +L+(U,Re)f+ +m+
+p( f+)
+
Z t
0
uf+
t +
ZD
J
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BGJ1*D
Integrate by parts over time and spaceZ0
ZD
u
t +L(U,Re)u+p
f+ + ( u)m+
=
Z t
0
ZD
u
f+
t +L+(U,Re)f+ +m+
+p( f+)
+
Z t
0
uf+
t +
ZD
J
Definition of adjoint problem
Assume volume forcing, mass source and integrate in timeu
t+L(U,Re)u+p= f u = Q
u(t) f+(t) u(0) f+(0) =
Z t
0
ZD
f f+ + Qm+
+
ZD
J n
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BGJ1*D
u(t) f+(t) u(0) f
+(0) =Z t0
ZD
f f
+ + Qm+
+ZD
J n
Assume initial condition for adjoint system f+
(t) =u(t)
u(t) u(t) =u(0) f+(0) +
Z t
0
ZD
f f+ + Qm+
+
ZD
J n
E ) ] ?) E
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u(t) u(t) =u(0) f+(0) +
Z t
0
ZD
f f+ + Qm+
+
ZD
J n
BGJ1*D
u(t) f+(t) u(0) f
+(0) =Z t0
ZD
f f
+ + Qm+
+ZD
J n
Assume initial condition for adjoint system f+
(t) =u(t)
djoint velocity gives sensitivity to initial condition and forcing
u2(t)
u(0)
=f+(0) u2(t)
f
=f+
E ) ] ?) E
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u(t) u(t) =u(0) f+(0) +
Z t
0
ZD
f f+ + Qm+
+
ZD
J n
BGJ1*D
u(t) f+(t) u(0) f
+(0) =Z t0
ZD
f f
+ + Qm+
+ZD
J n
Assume initial condition for adjoint system f+
(t) =u(t)
djoint pressure gives sensitivity to mass source
u2(t)
Q
=m+
E )L 1 ] ?) E G 1
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u(t) u(t) =u(0) f+(0) +
Z t
0
ZD
f f+ + Qm+
+
ZD
J n
BGJ1*D
u(t) f+(t)
u(0) f
+(0) =Z t0
ZD
f f
+ + Qm+
+ZD
J n
Assume initial condition for adjoint system f+
(t) =u(t)
Gradient of adjoint field gives sensitivity to boundary
conditions
u2(t)
uwall=
1
Ref+ +m+ n
4 ?1
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4"*?1(-
.*$C1?1*D @E F"1)
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.*'"#*"'$? $'$0-*-'QI-D(@?)< ("0C-'7 C$
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.*'"#*"'$? $'$0-*-'QI-D(@?)< ("0C-'7 C$
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.*'"#*"'$? $'$0-*-'QI-D(@?)< ("0C-'7 C$
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.*'"#*"'$?
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.*'"#*"'$?
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.-(
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.-(
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3?@N #/$'* E@'
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3?@N $'@"() $ #D?1()-'
XO C$
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3?@N $'@"() $ #D?1()-'
XO C$
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3?@N $'@"() $ #D?1()-'
XO C$
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3?@N $'@"() $ #D?1()-'
XO C$
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3?@N $'@"() $ #D?1()-'
XO C$
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3?@N $'@"() $ #D?1()-'
XO C$
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4-( F@N
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D D
Oscillators !Modal analysis
M !$'A-G0$? E@'#1(A R>
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^?@C$??D "(>'@$#/ 1)-(G]-< $-#1]# $G$? >@
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!$A'$(A- 1)-(G*D $()
$)L@1(* -_"$G@( AD >
./-))1(A 0@)- =Q N$J-0$U-' J
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6@(-'$*@' -H>'-
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6@(-'$*@' -H>'-
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.*'"#*"'$? J$'1$G@(< @E */- C$
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5$U- #@(*'@? CD 0-$(< @E
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;@)- =
.-(
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;@)- =
.-(
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;@)- =
9@*$? .-(
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;@)- ==
.-(
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;@)- ==
.-(
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;@)- ==
9@*$? .-(
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.10"?$G@(< @E $ @
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P@1?1]-''-G0$? '-@(?1]-'
&'$()* -* $?O7 `@"'($? @E 3?"1) ;-#/O7 [\XX
^'$)1-(* @E */- '-
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4>G0$? E@'#1(A $() '-@( @E @>G0$? E@'#1(A $() '-@(
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F247%" 1#)9,*.G ?,H@I2 -J-9(
#.*'-$0N1?$*-
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F247%" 1#)9,*.G KA@/%:-+
#6@(J-#GJ- (@(M(@'0$?1*D7 [h E@'#1(A $* E'-_"-(#D 3cX\\O
#3@'#1(A $#GJ- C-*N--( C'$(#/ = $() ==
Brandt et al, JFM, 2011
.-(
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9.MN$J-
^-(-'$?1W- *@ G0-M>-'1@)1# $() G0-M)->-()-(* F@N< I-?$*- (@(M0@)$? $($?DG01W$G@( >'@C?-0"*M@"*>"* $($?D
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> > D
.--U @>G0$? -(-'AD $0>?1]#$G@(
G(t) = maxq0
hq, qi
hq0, q0i
=(>"*M@"*>"* $($?D
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> > D
.--U @>G0$? -(-'AD $0>?1]#$G@(
G(t) = maxq0
hq, qi
hq0, q0i
= maxq0
hA(t)q0, A(t)q0ihq0, q0i
=(>"*M@"*>"* $($?D
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> > D
.--U @>G0$? -(-'AD $0>?1]#$G@(
G(t) = maxq0
hq, qi
hq0, q0i
= maxq0
hA(t)q0, A(t)q0ihq0, q0i
= maxq0
hAH(t)A(t)q0, q0i
hq0, q0i
=(>"*M@"*>"* $($?D
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E@' G0-M)->-()-(* "* $($?D
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E@' G0-M)->-()-(* '@C?-0
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E@' G0-M)->-()-(* '@C?-0
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$)L@1(*
>'@C?-0
(n)
)1'-#*>'@C?-0
'@C?-0
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g$'1$G@($? E@'0"?$G@( @E */- @>G0$? A'@N*/ >'@C?-0more general
J =
||q||2
||q0||2 maxWe wish to maximize
with the constraintd
dt
q=L(t)q
Listen Carlo Cossu on Friday!
^?@C$? 0@)-
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;@'@C?-0< N1*/ 1(/@0@A-(-@"< )1'-#G@(?-H A-@0-*'D
6$((@* "G0$?-() @( 0@'- */$(
@(- )1'-#G@( u+(x, y)
6@0>"*$G@($? 1
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One vs two inhomogeneous directions:state vector, matrix, operation count
L CN
2N
2
O(N4)L CNN
O(N2)
O(N3) O(N6)
Matrix size
Operation count
State vectorq=
q1
q2
qN
q=
q1,1
q1,2
qN,N
6@0>"*$G@($? 1
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One vs two inhomogeneous directions:state vector, matrix, operation count
" Direct eigenvalue algorithms become too expensive
"
Iterative algorithms,Arnoldi technique
L CN
2N
2
O(N4) O(N6)
storage
CPU time
K'(@?)1 $?A@'1*/0
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K#G@( @E */- ?1(-$' @>-'$*@' N1*/1( $( @'*/@(@'0$? C$
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K#G@( @E */- ?1(-$' @>-'$*@' N1*/1( $( @'*/@(@'0$? C$'-
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Only multiplication by L are necessary
System eigenvalues approximated
by eigenvalues of H
Eig(L) Eig(H)
mH$0>?- 1( [hQ F@N 1( sML"(#G@(
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Base flow
Lashgari et al, in preparation
mH$0>?- 1( [hQ F@N 1( sML"(#G@(
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Spectrum
Lashgari et al, in preparation
" Steady two-dimensional bifurcation
mH$0>?- 1( [hQ F@N 1( sML"(#G@(
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Eigenfunction
Lashgari et al, in preparation
"
Steady two-dimensional bifurcation
mH$0>?- 1( [hQ F@N 1( sML"(#G@(
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New asymmetric state
Lashgari et al, in preparation
"
Steady two-dimensional bifurcation
sML"(#G@(Q mH$0>?- 1( [7iha
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Lashgari et al, in preparation
Asymmetric state unstable to 3D periodic disturbances
Snapshot method based on linear DNS
mH$0>?- 1( ZhQ `-* 1( #'@
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147/172
Bagheri et al, JFM, 2011, Schlatter et al.
Use DNS and compute spectrum of matrix exponentialq= exp(tL)q0
."00$'D
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.*$C1?1*D @E F"1) '@C?-0-()-(* >'@C?-0Q -1A-(J$?"- >'@C?-0 1( G0-O mHb
6?$
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,$'$??-?
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4''M.@00-'E-?) $() ._"1'- -_"$G@(
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P@(0@)$?
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=(>"*M@"*>"* $>>'@$#/
P@((@'0$? @>-'$*@'